3.7.81 \(\int \frac {(d+e x)^2}{(d^2-e^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=53 \[ \frac {x}{3 d^2 \sqrt {d^2-e^2 x^2}}+\frac {2 (d+e x)}{3 e \left (d^2-e^2 x^2\right )^{3/2}} \]

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Rubi [A]  time = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {653, 191} \begin {gather*} \frac {x}{3 d^2 \sqrt {d^2-e^2 x^2}}+\frac {2 (d+e x)}{3 e \left (d^2-e^2 x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2/(d^2 - e^2*x^2)^(5/2),x]

[Out]

(2*(d + e*x))/(3*e*(d^2 - e^2*x^2)^(3/2)) + x/(3*d^2*Sqrt[d^2 - e^2*x^2])

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 653

Int[((d_) + (e_.)*(x_))^2*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)*(a + c*x^2)^(p + 1))/(c*(
p + 1)), x] - Dist[(e^2*(p + 2))/(c*(p + 1)), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, p}, x] &&
EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && LtQ[p, -1]

Rubi steps

\begin {align*} \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx &=\frac {2 (d+e x)}{3 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{3} \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx\\ &=\frac {2 (d+e x)}{3 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x}{3 d^2 \sqrt {d^2-e^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 47, normalized size = 0.89 \begin {gather*} \frac {(2 d-e x) (d+e x)}{3 d^2 e (d-e x) \sqrt {d^2-e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^2/(d^2 - e^2*x^2)^(5/2),x]

[Out]

((2*d - e*x)*(d + e*x))/(3*d^2*e*(d - e*x)*Sqrt[d^2 - e^2*x^2])

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IntegrateAlgebraic [A]  time = 0.37, size = 42, normalized size = 0.79 \begin {gather*} \frac {(2 d-e x) \sqrt {d^2-e^2 x^2}}{3 d^2 e (d-e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d + e*x)^2/(d^2 - e^2*x^2)^(5/2),x]

[Out]

((2*d - e*x)*Sqrt[d^2 - e^2*x^2])/(3*d^2*e*(d - e*x)^2)

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fricas [A]  time = 0.40, size = 71, normalized size = 1.34 \begin {gather*} \frac {2 \, e^{2} x^{2} - 4 \, d e x + 2 \, d^{2} - \sqrt {-e^{2} x^{2} + d^{2}} {\left (e x - 2 \, d\right )}}{3 \, {\left (d^{2} e^{3} x^{2} - 2 \, d^{3} e^{2} x + d^{4} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2/(-e^2*x^2+d^2)^(5/2),x, algorithm="fricas")

[Out]

1/3*(2*e^2*x^2 - 4*d*e*x + 2*d^2 - sqrt(-e^2*x^2 + d^2)*(e*x - 2*d))/(d^2*e^3*x^2 - 2*d^3*e^2*x + d^4*e)

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giac [A]  time = 0.27, size = 48, normalized size = 0.91 \begin {gather*} -\frac {\sqrt {-x^{2} e^{2} + d^{2}} {\left (x {\left (\frac {x^{2} e^{2}}{d^{2}} - 3\right )} - 2 \, d e^{\left (-1\right )}\right )}}{3 \, {\left (x^{2} e^{2} - d^{2}\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2/(-e^2*x^2+d^2)^(5/2),x, algorithm="giac")

[Out]

-1/3*sqrt(-x^2*e^2 + d^2)*(x*(x^2*e^2/d^2 - 3) - 2*d*e^(-1))/(x^2*e^2 - d^2)^2

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maple [A]  time = 0.05, size = 44, normalized size = 0.83 \begin {gather*} \frac {\left (e x +d \right )^{3} \left (-e x +d \right ) \left (-e x +2 d \right )}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d^{2} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2/(-e^2*x^2+d^2)^(5/2),x)

[Out]

1/3*(e*x+d)^3*(-e*x+d)*(-e*x+2*d)/d^2/e/(-e^2*x^2+d^2)^(5/2)

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maxima [A]  time = 1.38, size = 58, normalized size = 1.09 \begin {gather*} \frac {2 \, x}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}} + \frac {2 \, d}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e} + \frac {x}{3 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2/(-e^2*x^2+d^2)^(5/2),x, algorithm="maxima")

[Out]

2/3*x/(-e^2*x^2 + d^2)^(3/2) + 2/3*d/((-e^2*x^2 + d^2)^(3/2)*e) + 1/3*x/(sqrt(-e^2*x^2 + d^2)*d^2)

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mupad [B]  time = 0.48, size = 38, normalized size = 0.72 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (2\,d-e\,x\right )}{3\,d^2\,e\,{\left (d-e\,x\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^2/(d^2 - e^2*x^2)^(5/2),x)

[Out]

((d^2 - e^2*x^2)^(1/2)*(2*d - e*x))/(3*d^2*e*(d - e*x)^2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{2}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**2/(-e**2*x**2+d**2)**(5/2),x)

[Out]

Integral((d + e*x)**2/(-(-d + e*x)*(d + e*x))**(5/2), x)

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